Entropic Order Parameters for the Phases of QFT

We start by describing how generalized symmetries in QFT arise in the violation of elementary properties that appear when we associate algebras to regions in QFT. This observation provides a new perspective/proof of the abelian nature of generalized symmetries. Further, the algebra of order/disorder parameters is fixed and can be deduced without the explicit construction of the non-local operators. In these scenarios, there are two natural algebras associated with regions of specific topologies, suggesting a simple geometrical order parameter defined as a relative entropy. These relative entropies satisfy a “certainty relation” connecting the statistics of the order and disorder parameters. We describe old and new aspects of the phases of QFT’s with generalized symmetries from this perspective. In particular, the certainty relation makes transparent the duality between constant and area law behaviors in symmetry-breaking scenarios, and in CFT’s there are relative entropies exactly computable.